
\begin{section}{Discussion}
\label{sec:con_disc}
 
In this thesis we have investigated database design in different data models: the relational model, XML documents and Description Logic Knowledge Bases. One important goal of database design is to avoid redundancies arising from badly designed models. In the relational model we have focused on FDs. Normal Forms, especially Boyce-Codd Normal Form, avoid redundancies arising from FDs. For XML documents we have summarized the work by Arenas and Libkin \cite{2004arenas}. They have introduced XFDs and XML Normal Form. \\

\noindent As the data available in graph databases, especially in the Semantic Web, grows, it is needed to focus on consistency and redundancy in such data models. Therefore, data design must also play an important role in graph databases. We have used ideas from the work on normal forms in the relational model and XML documents in order to find a normal form for graph databases. \\

\noindent First, we have fixed $\DLA$ as a formal model for graph databases. We introduced the relational to Description Logic direct mapping for translating relational schemas to $\DLA$ KBs. We considered path-based identification constraints \cite{2008calvanese} as a formalism to model FDs in $\DLA$ KBs. We then showed that the direct-mapping extended with pIdCs is not semantics preserving. Therefore, we extended pIdCs and introduced tree-based identification constraints. We showed that the direct-mapping extended with tIdCs is indeed semantics preserving. \\

\noindent Tree-based identification constraints allowed us to introduce Description Logic Normal Form (k-DLNF). DLNF tries to avoid redundancies in DL KBs analogously to BCNF in relational databases. As we have shown in Section \ref{sec:dl_nf}, a relational schema is in BCNF if and only if the $\DLA$ KB, translated from this relational schema, is in 2-DLNF.
\end{section}

\begin{section}{Future Work}
\label{sec:con_future} 
 As an extension of this work in the future we will focus on at least three major topics:
 \begin{itemize}
  \item First, it is needed to thoroughly investigate further \emph{properties of tree-based identification constraints}. We have given an algorithm to decide the implication problem for tIdCs over weakly-acyclic KBs. It remains to show, if there is an algorithm for the implication problem of tIdCs over arbitrary KBs. The same also holds for pIdCs. Another open question is, whether there exists an inference system, similar to Armstrong Axioms, for pIdCs.
  \item Second, we need to \emph{extend the theory for DLNF}. We have defined DLNF as a faithful translation of BCNF to DLs. The most important question is, if there is an algorithm that efficiently checks if a $\DLA$ KB is in DLNF. This involves the computation of the closure of tIdCs, which is also a problem open to be solved. Furthermore, it is interesting to look for a decomposition algorithm that repairs $\DLA$ KBs which are not in DLNF.
  \item In this work we have extensively studied the relationship between the relational model and DL KBs. It remains to study also \emph{the relationship between XML documents and DL KBs}. Therefore, it would be interesting to find a direct-mapping from XML documents to $\DLA$ KBs.  We can then ask if it also holds that whenever an XML document is in XNF, the $\DLA$ KB, translated from this XML document, is in DLNF and vice versa.
 \end{itemize}

\end{section}

